Question

How do you solve `0.23x^2+6.5x+4.3<0` using a sign chart?

Answer 1

Solution : `-27.58 < x < -0.68 or x| (-27.58,-0.68)`

Explanation:

`0.23x^2+6.5x+4.3<0`

Comparing with standard quadratic equation `ax^2+bx+c=0`

`a= 0.23 ,b=6.5 ,c=4.3` Discriminant `D= b^2-4ac` or

`D ~~ 38.29` If discriminant positive, we get two real solutions,

Quadratic formula: `x= (-b+-sqrtD)/(2a)`or

`x= (-6.5+-sqrt38.29)/(2*0.23) :. x ~~ -27.58 , x ~~ -0.68`

`0.23x^2+6.5x+4.3<0`or

`f(x)=0.23 (x +27.58)(x+0.68) <0` .

Critical points are `x ~~ -27.58 , x ~~ -0.68`

Sign chart: When `x< -27.58` sign of `f(x)` is `(-) * (-) = (+) ; > 0`

When `-27.58 < x < -0.68` sign of `f(x)` is `(+) * (-) = (-) ; < 0`

When `x > -0.68` sign of `f(x)` is `(+) * (+) = (+) ; > 0`

Solution : `-27.58 < x < -0.68 or x| (-27.58,-0.68)`

graph{0.23x^2+6.5x+4.3 [-160, 160, -80, 80]}

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